Natural Oscillations of a Simply Supported Square Plate

Objective: Modal analysis of a simply supported square plate.

Initial data file: 5_2.spr

Problem formulation: Determine the natural oscillation modes and frequencies ω of the simply supported square plate with the density of the material ρ.

References: I.A. Birger, Ya.G. Panovko, Strength, Stability, Vibrations, Handbook in three volumes, Volume 3, Moscow, Mechanical engineering, 1968, p. 375.

Initial data:

E = 2.06·108 kPa - elastic modulus;
ν = 0.3 - Poisson’s ratio;
ρ = 7.85 t/m3 - density of the material;
h = 0.01 m - thickness of the plate;
a1 = 1.0 m - long side of the plate (along the X axis of the global coordinate system);
a2 = 1.0 m - short side of the plate (along the Y axis of the global coordinate system).

 

Finite element model: Design model – grade beam / plate, 400 plate elements of type 20. The spacing of the finite element mesh along the sides of the plate (along the X, Y axes of the global coordinate system) is 0.05 m. Boundary conditions are provided by imposing constraints in the direction of the degree of freedom Z for the edges parallel to the X and Y axes of the global coordinate system. The distributed mass is specified by transforming the static load from the self-weight of the plate ow = γ∙h, where γ = ρ∙g = 77.01 kN/m3. Number of nodes in the design model – 441. The determination of the natural oscillation modes and natural frequencies is performed by the Lanczos method. A consistent mass matrix is used in the calculation.

 

 

Results in SCAD



Design model

 



1-st natural oscillation mode

 



2-nd natural oscillation mode

 



3-rd natural oscillation mode

 



4-th natural oscillation mode

 



5-th natural oscillation mode

 



6-th natural oscillation mode

 



7-th natural oscillation mode

 



8-th natural oscillation mode

 



9-th natural oscillation mode

 



10-th natural oscillation mode

 



11-th natural oscillation mode

 



12-th natural oscillation mode

 



13-th natural oscillation mode

 



14-th natural oscillation mode

 



15-th natural oscillation mode

 



16-th natural oscillation mode

 



17-th natural oscillation mode

 



18-th natural oscillation mode

 



19-th natural oscillation mode

 



20-th natural oscillation mode

 

 

Comparison of solutions:

Natural frequencies ω, rad / s

Oscillation mode

Number of half waves
m1, m2

Theory

SCAD

Deviations, %

1

1, 1

306.0

306,1

0,05

2

1, 2

765. 0

765,6

0,08

3

2, 1

765.0

765,6

0,08

4

2, 2

1224.0

1226,4

0,19

5

1, 3

1530.0

1531,4

0,09

6

3, 1

1530.0

1531,4

0,09

7

2, 3

1989.0

1994,4

0,27

8

3, 2

1989.0

1994,4

0,27

9

1, 4

2601.0

2603,6

0,10

10

4, 1

2601.0

2603,6

0,10

11

3, 3

2754.0

2766,2

0,44

12

2, 4

3060.0

3069,7

0,32

13

4, 2

3060.0

3069,7

0,32

14

3, 4

3825.0

3846,8

0,57

15

4, 3

3825.0

3846,8

0,57

16

1, 5

3978.0

3982,6

0,12

17

5, 1

3978.0

3982,6

0,12

18

2, 5

4437.0

4452,7

0,35

19

5, 2

4437.0

4452,7

0,35

20

4, 4

4896.0

4934,7

0,79

 

 

Notes: In the analytical solution the natural frequencies ω of the simply supported square plate with the density of the material ρ can be determined according to the following formula:

\[\omega =\pi^{2}\cdot \left( {\frac{m_{1}^{2}}{a_{2}^{2}}+\frac{m_{2} ^{2}}{a_{2}^{2}}} \right)\cdot \left( {\frac{D}{\rho \cdot h}} \right)^{\frac{1}{2}},\quad where:\quad D=\frac{E\cdot h^{3}}{12\cdot \left( {1-\mu ^{2}} \right)}\quad, \quad m_{1}, m_{2} =1,2,3, ... \]